We start this discussion by a heuristic, TBA-inspired, approach to soliton gases. More explicitly, we follow the classical TBA approach, already discussed in the context of Toda gas in [1]}, as a way to heuristically recover the dispersion relation (REF ), analogous to the thermodynamic Bethe ansatz equation (REF ), albeit in the context of a soliton gas. The aim of this section is to build intuition and further reinforce the aforementioned parallels between soliton gas theory and GHD. It will also serve as a vector to naturally introduce notions such as the “asymptotic space”, and a geometric interpretation of the classical TBA which will provide much insight into the hydrodynamics and thermodynamics of the soliton gas.
